f(x)=(x^(2)-2)e^(-2x) how do i show that f'(x)= -2e^-2x(x+1)(x-2) and f''(x)=2e^-2x(2x^2-4x-3)

Answer 1

Since f(x) = (x² -2)(e^-2x), then f(x) is the product of two functions so to take the derivative requires us to use the product rule. The product rule goes like this: for f(x) = u•v, then f '(x) = u'v + uv'

u = x² -2

u' = 2x

v = e^-2x

v' = -2e^-2x

f '(x) = (2x)(e^-2x) + (x² -2)(-2e^-2x)

factor out -2e^-2x to get:

f '(x) = -2e^-2x(-x + (x² - 2))

f '(x) = -2e^-2x(x² - x - 2)

f '(x) = -2e^-2x(x - 2)(x + 1)

To find f ''(x), I would start with f '(x) = -2e^-2x(x² - x - 2) just so I didn't need to use the product rule twice:

u = -2e^-2x

u' = 4e^-2x

v = x² - x - 2

v' = 2x - 1

f ''(x) = 4e^-2x(x² - x - 2) + (-2e^-2x)(2x - 1)

Then factor out 2e^-2x to get:

f ''(x) = 2e^-2x[(2(x² - x - 2)) + (-1)(2x - 1)]

f ''(x) = 2e^-2x[(2x² - 2x - 4) + (-2x + 1)]

f ''(x) = 2e^-2x[2x² - 2x - 4 - 2x + 1]

f ''(x) = 2e^-2x(2x² - 4x - 3)